534? Royal Irish Academy. 



The method of coordinates here employed by Mr. Graves is entirely 

 distinct from that which is developed by Mr. Davies in a paper in 

 the 12th vol. of the Transactions of the Royal Society of Edinburgh. 

 Mr. Graves apprehends, however, that he has been anticipated in the 

 choice of these coordinates by M. Gudermann of Cleves, who is the 

 author of an " Outline of Analytic Spherics," which Mr. Graves 

 has been unable to procure. 



The President communicated a new demonstration of Fourier's 

 theorem. 



A letter was read from Professor Holmboe, accompanying his me- 

 moir, De Prised Re Monetarid Norvegia, &c, and requesting to know 

 from the Academy whether any of the coins described in that work 

 are found in Ireland*. 



July 12f. — Part I. of a " Memoir on the Dialytic Method of Eli- 

 mination," by J. J. Sylvester, Esq., A.M., of Trinity College, Dublin, 

 and Professor of Natural Philosophy in University College, London, 

 was read. 



The author confines himself in this part to the treatment of two 

 equations, the final and other derivees of which form the subject of 

 investigation. 



The author was led to reconsider his former labours in this de- 

 partment of the general theory by finding certain results announced 

 by M. Cauchy in L'Institut, March Number of the present year, 

 which flow as obvious and immediate consequences from Mr. Syl- 

 vester's own previously published principles and method. 



Let there be two equations in x, 



U = a x n + b x 11 - 1 + c x n ~ 2 + e x n ~ 3 + &c. = 0, 



V=ax w +|3/- 1 + X^- 2 + &c. =0, 



and let n = m + i, where ; is zero or any positive value (as may be). 

 Let any such quantities as x r U, x e V, be termed augmentatives 

 of U or V. 



To obtain the derivee of a degree s units lower than V, we must 

 join s augmentatives of U with s -f < of V. Then out of 2 s-)- i 

 equations 



x° . U = 0, x\ . U = 0, * 2 . U = 0, X s - 1 . U = 0, 



x°.V = 0, x>.V = 0, *°-.V = 0, ^+*- 1 .v = o, 



we may eliminate linearly 2 s -f- i — 1 quantities. 



Now these equations contain no power of x higher than 

 m _j_ i _j. 5 — 1 ; accordingly, all powers of x, superior torn — s, may 

 be eliminated, and the derivee of the degree (m — s) obtained in its 

 prime form. 



Thus to obtain the final derivee (which is the derivee of the de- 

 gree zero), we take m augmentatives of U with n of V, and elimi- 

 nate (m + n — 1) quantities, namely, 



x, x 2 , x*, up to x m + n ~ 1 . 



* The Committee of Antiquities, having been consulted on this point, 

 reported in the negative. 



[f An abstract of Prof. Lloyd's paper read on June 14th, will be found 

 in the present volume, p. 395.] 





